On the other hand, Ref. [1]} presented the first MRT-LB formulation on a two-dimensional nine velocities (D2Q9) rectangular lattice grid by introducing coupling between various relaxation parameters and the grid aspect ratio via a linear stability analysis. However, as shown later in Ref. [2]}, this scheme is not able to completely recover the isotropy of the macroscopic fluid flow equations. A different rectangular MRT-LB approach which maintains the transformation matrix independent of the grid aspect ratio [3]} was found to exhibit similar spurious behavior. More recently, via an inverse design analysis based on the Chapman-Enskog expansion [4]}, Ref. [2]} introduced a rectangular MRT-LB method with an additional adjustable parameter that determines the relative orientation in the energy-normal stress subspace, which can be adjusted to completely eliminate the anisotropy. However, the resulting scheme appears to be quite complicated in specifying such an additional parameter as a function of the speed of sound and the grid aspect ratio, and with stable results achieved only for the grid aspect ratio above 0.2. Later, inspired by the lattice kinetic scheme [6]}, Ref. [7]} presented a consistent MRT-LBM on a rectangular grid in which the equilibrium moments are extended to include the stress components, which are designed in such a way as to restore the isotropy of the recovered hydrodynamical equations. However, the guidance for setting up the associated free parameters to recover the physically correct transport coefficients seems involved. While this scheme showed good agreement with benchmark results, results on its numerical stability at relatively low viscosities or large Reynolds numbers at different grid aspect ratios were not reported. Moreover, all the existing MRT-LB schemes on rectangular lattice grids involve raw moments, where the moment basis are orthogonalized via a Gram-Schmidt orthogonalization; however, it has recently been demonstrated that the orthogonalization can couple the evolution of the higher order moments to those of the lower moments thereby impacting their numerical stability characteristics [8]}. In the context of the rectangular lattice grid, the use of such an orthogonal moment basis also results in unwieldy expressions for the transformation matrices dependent on the lattice grid ratio, which compromises their implementation. Moreover, the existing rectangular LB schemes do not eliminate the cubic velocity errors arising from aliasing effects on the D2Q9 lattice.
